Some uses of homogeneous forcing David Asper´ o University of East Anglia Torino, 27–05–2015
A forcing notion P is homogeneous iff for all p , p 0 2 P there are q P p and q 0 P p 0 such that P � q ⇠ = P � q 0 Standard fact: If P is a homogeneous forcing notion, then for all p , p 0 2 P and all statements ' in the forcing language for P with parameters from the ground model, p � P ' iff p 0 � P '
Hilbert’s programme revisited Some local notation : Given a theory Σ and a sentence � , in the language of set theory, � is a Φ –consequence from Σ , denoted Σ ` Φ � , iff for every set–forcing P , if P forces every sentence in Σ , then P forces � . Φ is for ‘forcing’. This definition of course makes sense for choices of Σ for which this can be expressed. For choices of Σ where its members have unbounded L´ evy complexity this might of course not be definable. Also, note that the definition makes sense also for choices of Σ which are not even definable (as long as they are in V ).
This gives a notion of logic | = Φ , possibly weaker than the logic | = GM of the generic multiverse. We use Φ –true, Φ –satisfiable, Φ –complete and so on, in the natural intended way. For example, a theory Σ is Φ –complete for a set ∆ of sentences if and only if for every � 2 ∆ at least one of Γ | = Φ � and Γ | = Φ ¬ � holds. The usual (Woodin’s) definition of Ω –logic can be phrased in the above language, at least for (say) choices of Σ which are definable over ! : Suppose Σ is definable over ! . Then � is an Ω –consequence of Σ if and only if the sentence “for all ordinals ↵ , if V α | = for every 2 Σ , then V α | = � ” is a Φ –truth (where of course the mention of Σ refers to the definition of Σ ).
We may also define relativized versions Φ Γ of Φ –logic for definable classes Γ of posets. For example T is Φ Γ –complete for ∆ iff for every � 2 ∆ it holds that either • for every P 2 Γ , if � P ' for every ' 2 T , then � P � , or • for every P 2 Γ , if � P ' for every ' 2 T , then � P ¬ � .
Σ 2 theories For Σ 2 theories, i.e., theories of the form ( 9 ↵ )( V α | = T ) (equivalently, of the form ( 9 )( H ( ) | = T ) ) and Σ 2 sentences � , Φ –logic coincides with Ω –logic: T | = Φ � iff T | = Ω �
Woodin: If there is a proper class of Woodin and the Ω Conjecture is true, then: 1 the P m ax –axiom ( ⇤ ) is Ω –satisfiable (equiv., it can always be obtained by set–forcing over any set–forcing extension). Hence, since ( ⇤ ) is Φ –complete for Th ( H ( ! 2 )) , if the Ω Conjecture is true under every large cardinal hypothesis, then ( ⇤ ) is an axiom which is • compatible with all large cardinals, • Ω –complete for Th ( H ( ! 2 )) , and • which can always be set–forced after any set–forcing. 2 There is no Ω –satisfiable theory which is Ω –complete for Th ( H ( � + 0 )) , where � 0 is the least Woodin cardinal. Woodin: If there is a proper class of Woodin and the Strong Ω Conjecture is true, then: 1 The Ω Conjecture is true. 2 All theories which are Ω –complete for Th (( H ( ! 2 )) imply ¬ CH. 3 There is no Ω –satisfiable theory which is Ω –complete for the Σ 2 3 theory.
In “Incompatible Ω –complete theories”, JSL 2009, Koellner and Woodin contemplate the following very optimistic scenario: Could it be, in a large cardinal context, that the following holds? (i) The Ω Conjecture is false. (ii) There is a sequence of Ω –satisfiable Σ 2 theories which are Ω –complete for the theory of larger and larger (all ?) reasonably specifiable initial segments of the universe. (iii) All these theories give the same theory of the relevant initial segments of the universe. Koellner and Woodin show that if (i) and (ii) hold, then (iii) has to fail (granting liberal use of large cardinals, as usual).
In “Incompatible Ω –complete theories”, JSL 2009, Koellner and Woodin contemplate the following very optimistic scenario: Could it be, in a large cardinal context, that the following holds? (i) The Ω Conjecture is false. (ii) There is a sequence of Ω –satisfiable Σ 2 theories which are Ω –complete for the theory of larger and larger (all ?) reasonably specifiable initial segments of the universe. (iii) All these theories give the same theory of the relevant initial segments of the universe. Koellner and Woodin show that if (i) and (ii) hold, then (iii) has to fail (granting liberal use of large cardinals, as usual).
They show that if there is a Σ 2 theory T which, modulo some large cardinal assumption LC, is Ω –satisfiable and Ω –complete for (say) Th ( H ( )) , for = ( 2 @ 0 ) + , then there are Σ 2 theories T CH , T ¬ CH which, modulo slightly stronger large cardinal assumption LC 0 , are Ω –satisfiable and Ω –complete for Th ( H ( ! 2 )) and such that • T CH ` CH and • T ¬ CH ` ¬ CH.
Proof proceeds by considering the theories that (essentially) say � “I am a forcing extension of a model of T by Add ( ! 1 , 1 ) ” (for T CH ) � “I am a forcing extension of a model of T by Add ( ! , ! 2 ) ” (for T ¬ CH ) The main points are: • Add ( ! 1 , 1 ) and Add ( ! , ! 2 ) are definable over H ( ) from no parameters and homogeneous. • is large enough that all nice names for members of H ( ! 2 ) are in H ( ) . CH, ¬ CH is clearly not the only pair they can deal with. A similar result can be proved for any Σ 2 statement � such that both � and ¬ � can be forced by some similarly nice forcing.
Proof proceeds by considering the theories that (essentially) say � “I am a forcing extension of a model of T by Add ( ! 1 , 1 ) ” (for T CH ) � “I am a forcing extension of a model of T by Add ( ! , ! 2 ) ” (for T ¬ CH ) The main points are: • Add ( ! 1 , 1 ) and Add ( ! , ! 2 ) are definable over H ( ) from no parameters and homogeneous. • is large enough that all nice names for members of H ( ! 2 ) are in H ( ) . CH, ¬ CH is clearly not the only pair they can deal with. A similar result can be proved for any Σ 2 statement � such that both � and ¬ � can be forced by some similarly nice forcing.
Down to H ( ω 2 ) Consider the question: Question: Does the existence of an Ω –satisfiable Σ 2 –theory T which is Ω –complete for Th ( H ( ! 2 )) imply the existence of another such theory incompatible with T ? [Koellner–Woodin] does not address this question: their use of Add ( ! , ! 2 ) does address the problem of producing a theory implying ¬ CH, but Add ( ! 1 , 1 ) is not suitable for building a theory implying CH (in our context): If CH fails, then there are nice Add ( ! 1 , 1 ) –names for members of H ( ! 2 ) which are not in H ( ! 2 ) . In fact Add ( ! 1 , 1 ) will collapse ! 2 .
Down to H ( ω 2 ) Consider the question: Question: Does the existence of an Ω –satisfiable Σ 2 –theory T which is Ω –complete for Th ( H ( ! 2 )) imply the existence of another such theory incompatible with T ? [Koellner–Woodin] does not address this question: their use of Add ( ! , ! 2 ) does address the problem of producing a theory implying ¬ CH, but Add ( ! 1 , 1 ) is not suitable for building a theory implying CH (in our context): If CH fails, then there are nice Add ( ! 1 , 1 ) –names for members of H ( ! 2 ) which are not in H ( ! 2 ) . In fact Add ( ! 1 , 1 ) will collapse ! 2 .
Addressing the question Plan : Use [Koellner–Woodin]’s result in the following form: Theorem [Koellner–Woodin] Suppose there is a proper class of Woodin cardinals. Suppose ' is a Σ 2 large cardinal property and is a Σ 2 sentence such that T = ZFC + + “There is a proper class of Wodin cardinals” + “There is a proper class of ' –cardinals” is Ω –complete for Th ( H ( ! 2 )) . Let P ✓ H ( ! 2 ) be a forcing such that T Ω –implies that (1) P is definable over H ( ! 2 ) (from no parameters). (2) P is homogeneous. (3) P preserves ! 1 and has the @ 2 –c.c. (in particular every P –name for a member of H ( ! 2 ) can be assumed to be in H ( ! 2 ) ).
Let T P be the sentence: There is ( , N , G ) such that • is an inaccessible cardinal, • N | = T , • G is P N –generic over H ( ! 2 ) N , and • H ( ! 2 ) = H ( ! 2 ) N [ G ] . Then the sentence ZFC+ T P +“There is a proper class of Wodin cardinals” + “There is a proper class of ' –cardinals” is Ω –complete for Th ( H ( ! 2 )) .
A definable homogeneous version of the Hechler iteration Goal: Want to force b > ! 1 by a forcing P ✓ H ( ! 2 ) such that: (1) P is definable over H ( ! 2 ) (from no parameters). (2) P is homogeneous. (3) P preserves ! 1 and has the @ 2 –c.c. (in particular every P –name for a member of H ( ! 2 ) can be assumed to be in H ( ! 2 ) ). (4) P forces b > ! 1 .
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